PERIODIC MOTION 325
250. Effect of Damping on the Period. — Substituting the values of a and 6 in the expression for the period,
where P0 is the period for the undamped motion. It is evident from equation (XXVI) that the damping increases the period.
VIBRATIONS ABOUT A POSITION OF EQUILIBRIUM.
251. Lagrange's Method.— In the various pendulum problems which we have discussed the vibrating body was considered to be either a particle or a rigid body. These simplifications were necessary because the methods we have used cannot be applied conveniently to complicated systems. La-grange (1736-1813) introduced into Dynamics a method which can be applied to any vibrating system. The following is a special case of his method adapted to conservative systems which have only one degree of freedom of motion.
Express the potential energy of the system as a function of a properly chosen* coordinate qy so that when expanded in ascending powers of q the first power of q does not appear. Then the potential energy takes the form
U= ft + &22 + ft?3 + • • • , (XXVII)
where ft, ft, etc., are constants. The constant ft can be
* It is shown in books on advanced Dynamics that such a choice is always possible.